Realizations of kinetic differential equations

Gheorghe Craciun; Matthew D. Johnston; Gábor Szederkényi; Elisa Tonello; János Tóth; Polly Y. Yu; Gheorghe Craciun; Matthew D. Johnston; Gábor Szederkényi; Elisa Tonello; János Tóth; Polly Y. Yu

doi:10.3934/mbe.2020046

Mathematical Biosciences and Engineering

2020, Volume 17, Issue 1: 862-892. doi: 10.3934/mbe.2020046

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Research article Special Issues

Realizations of kinetic differential equations

1.
Department of Mathematics, University of Wisconsin-Madison, Madison, WI 53706-1325, USA
2.
Department of Biomolecular Chemistry, University of Wisconsin-Madison, Madison, WI 53706, USA
3.
Department of Mathematics and Computer Science, Lawrence Technological University, Southfield, MI 48075, USA
4.
Faculty of Information Technology and Bionics, Pázmány Péter Catholic University, Budapest, Hungary
5.
Department of Mathematics and Computer Science, Freie Universität, Berlin, Germany
6.
Department of Analysis, Budapest University of Technology and Economics, Budapest, Hungary
7.
Laboratory for Chemical Kinetics, Eötvös Loránd University, Budapest, Hungary

Received: 17 July 2019 Accepted: 18 October 2019 Published: 06 November 2019

The induced kinetic differential equations of a reaction network endowed with mass action type kinetics is a system of polynomial differential equations. The problem studied here is: Given a system of polynomial differential equations, is it possible to find a network which induces these equations; in other words: is it possible to find a kinetic realization of this system of differential equations? If yes, can we find a network with some chemically relevant properties (implying also important dynamic consequences), such as reversibility, weak reversibility, zero deficiency, detailed balancing, complex balancing, mass conservation, etc.? The constructive answers presented to a series of questions of the above type are useful when fitting differential equations to datasets, or when trying to find out the dynamic behavior of the solutions of differential equations. It turns out that some of these results can be applied when trying to solve seemingly unrelated mathematical problems, like the existence of positive solutions to algebraic equations.
- kinetic equations,
- reversibility,
- weak reversibility,
- mass action kinetics,
- reaction networks,
- realizations
Citation: Gheorghe Craciun, Matthew D. Johnston, Gábor Szederkényi, Elisa Tonello, János Tóth, Polly Y. Yu. Realizations of kinetic differential equations[J]. Mathematical Biosciences and Engineering, 2020, 17(1): 862-892. doi: 10.3934/mbe.2020046

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Abstract

The induced kinetic differential equations of a reaction network endowed with mass action type kinetics is a system of polynomial differential equations. The problem studied here is: Given a system of polynomial differential equations, is it possible to find a network which induces these equations; in other words: is it possible to find a kinetic realization of this system of differential equations? If yes, can we find a network with some chemically relevant properties (implying also important dynamic consequences), such as reversibility, weak reversibility, zero deficiency, detailed balancing, complex balancing, mass conservation, etc.? The constructive answers presented to a series of questions of the above type are useful when fitting differential equations to datasets, or when trying to find out the dynamic behavior of the solutions of differential equations. It turns out that some of these results can be applied when trying to solve seemingly unrelated mathematical problems, like the existence of positive solutions to algebraic equations.

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